Find the equation of the plane parallel to two lines

[Sho Kano]

Homework Statement

Let A, B and C be distinct vectors in V3 with B and C non-parallel.
a. Find an equation for the plane containing both the line through A parallel to B and the line through A parallel to C.
b. Verify that the two lines actually lie in the plane.

Homework Equations

The Attempt at a Solution

I can't see any situation where a plane containing A can be parallel to a plane containing C if they are distinct vectors (distinct as in non-parallel)

 

[Mark44]

Homework Statement

Let A, B and C be distinct vectors in V3 with B and C non-parallel.

What is V3? Presumably it's a 3-dimensional vector space. If so, is it different from $\mathbb{R}^3$?

a. Find an equation for the plane containing both the line through A parallel to B and the line through A parallel to C.

This is a very strange description. The context here seems to be that A is a point, but the earlier description is that A is a vector. I would guess that what they're calling vector A is the vector $\vec{OA}$, with O being the origin and A being the endpoint of the vector. Otherwise this problem doesn't make much sense.

Was the problem statement originally in some other language?

b. Verify that the two lines actually lie in the plane.

Homework Equations

The Attempt at a Solution

I can't see any situation where a plane containing A can be parallel to a plane containing C if they are distinct vectors (distinct as in non-parallel)

 

[andrewkirk]

The question is not well expressed. Here is what I think they mean.

Let A,B,C be three points in $\mathbb R^3$ and denote the origin by $O$.

Let L1 be a line through A that is parallel to line segment $\bar{OB}$.
Let L2 be a line through A that is parallel to line segment $\bar{OC}$.

(b) Show that there is a plane that contains both L1 and L2; and
(a) find the equation of that plane.

 

[Ray Vickson]

Homework Statement

Let A, B and C be distinct vectors in V3 with B and C non-parallel.
a. Find an equation for the plane containing both the line through A parallel to B and the line through A parallel to C.
b. Verify that the two lines actually lie in the plane.

Homework Equations

The Attempt at a Solution

I can't see any situation where a plane containing A can be parallel to a plane containing C if they are distinct vectors (distinct as in non-parallel)

While the question is not absolutely clear, I think one interpretation of it is that we have two vectors $\vec{B}$ and $\vec{C}$ in a 3-dimensional space; these are, perhaps, like force vectors or velocity vectors, having tails and heads not necessarily at the origin. Assuming that is the case, make a copy $\vec{C}'$ of $\vec{C}$ but whose tail coincides with the tail of $\vec{B}$, so that $\vec{B}$ and $\vec{C}'$ emanate from the same point in space, but have different directions. There certainly IS a plane P that contains both vectors $\vec{B}$ and $\vec{C}'$ (that is, which contains the three points that lie at the two ends of these vectors). Any Plane P that is parallel to P will be parallel to both vectors $\vec{B}$ and $\vec{C}$ (the original $\vec{C}$), and now all you need to is figure out which such plane passes through the point A which lies at the end of the vector $\vec{A}$ (assuming that this last vector has its tail at the origin).

 

[Sho Kano]

Was the problem statement originally in some other language?

I copied the problem straight off the problem set; I'll try to clarify with the teacher today

 

[Sho Kano]

What is V3? Presumably it's a 3-dimensional vector space. If so, is it different from R3R3\mathbb{R}^3?

There's no difference, my teacher just likes to use V3 as a distinction from R3 (for whatever reason) for the first few days of the class

 

[Sho Kano]

OK so apparently, we are supposed to find a plane that has both B and C in it, and A intersecting it. It makes sense because a plane can be parallel to many vectors.
The equation of such a plane can be $\left( \left< x,\quad y,\quad z \right> -\overrightarrow { OA } \right) \cdot \overrightarrow { B } \times \overrightarrow { C } =0$ right?
Then I guess I just do part b by plugging in OA + tB and OA + tC.